"Introduction". In: Fiber-Optic Communication Systems

406 CHAPTER 9. SOLITON SYSTEMS

9.1.2 Bright Solitons

Consider the case of anomalous GVD by settings=−1 in Eq. (9.1.3). It is common
to introduceu=NUas a renormalized amplitude and write the NLS equation in its
canonical form with no free parameters as

i

∂u

∂ξ

+

1

2

∂^2 u

∂τ^2

+|u|^2 u= 0. (9.1.5)

This equation has been solved by the inverse scattering method [14]. Details of this
method are available in several books devoted to solitons [11]–[13]. The main result
can be summarized as follows. When an input pulse having an initial amplitude

u( 0 ,τ)=Nsech(τ) (9.1.6)

is launched into the fiber, its shape remains unchanged during propagation whenN= 1
but follows a periodic pattern for integer values ofN>1 such that the input shape is
recovered atξ=mπ/2, wheremis an integer.
An optical pulse whose parameters satisfy the conditionN=1 is called thefun-
damental soliton. Pulses corresponding to other integer values ofNare calledhigher-
order solitons. The parameterNrepresents the order of the soliton. By noting that
ξ=z/LD, the soliton periodz 0 , defined as the distance over which higher-order soli-
tons recover their original shape, is given by

z 0 =

π

2

LD=

π

2

T 02

|β 2 |

. (9.1.7)

Thesoliton period z 0 andsoliton order Nplay an important role in the theory of optical
solitons. Figure 9.1 shows the pulse evolution for the first-order (N=1) and third-order
(N=3) solitons over one soliton period by plotting the pulse intensity|u(ξ,τ)|^2 (top
row) and the frequency chirp (bottom row) defined as the time derivative of the soliton
phase. Only a fundamental soliton maintains its shape and remains chirp-free during
propagation inside optical fibers.
The solution corresponding to the fundamental soliton can be obtained by solving
Eq. (9.1.5) directly, without recourse to the inverse scattering method. The approach
consists of assuming that a solution of the form

u(ξ,τ)=V(τ)exp[iφ(ξ)] (9.1.8)

exists, whereVmust be independent ofξfor Eq. (9.1.8) to represent a fundamental
soliton that maintains its shape during propagation. The phaseφcan depend onξbut
is assumed to be time independent. When Eq. (9.1.8) is substituted in Eq. (9.1.5) and
the real and imaginary parts are separated, we obtain two real equations forVandφ.
These equations show thatφshould be of the formφ(ξ)=Kξ, whereKis a constant.
The functionV(τ)is then found to satisfy the nonlinear differential equation

d^2 V

dτ^2

= 2 V(K−V^2 ). (9.1.9)